The infinitude of square-free palindromes

TL;DR

This work resolves the long-standing question of whether there are infinitely many square-free palindromes in any fixed base $b\ge 2$ and provides an asymptotic count for such palindromes under a coprimality constraint. The authors develop a novel hybrid approach that merges a $p$-adic/Archimedean van der Corput framework with an equidistribution estimate of Tuxanidy–Panario and an elementary sieve-based argument of Cilleruelo–Luca–Shparlinski. By splitting the square-divisor analysis into ranges for the divisor $d$ and proving sharp bounds for the counting function $S_b(x,D)$ in each range, they obtain an explicit asymptotic for the square-free palindromes among arithmetic-progressions coprime to $b^3-b$. The combination of these techniques yields both the infinitude result and a quantitative density, highlighting a surprising independence between palindrome-structure and square-free behavior within the restricted residue class. This advances understanding of the multiplicative structure of palindromes and introduces tools likely useful for related problems on digits, primes, and exponential sums in base-$b$ representations.

Abstract

We settle an open problem regarding palindromes; that is, positive integers which are the same when written forwards and backwards. In particular, we prove that for any fixed base $b\geq 2$, there exist infinitely many square-free palindromes in base $b$. We also provide an asymptotic expression for the number of such integers $\leq x$. The core of our proof utilises a hybrid $p$-adic/Archimedean van der Corput process, used in conjunction with an equidistribution estimate of Tuxanidy and Panario, as well as an elementary argument of Cilleruelo, Luca and Shparlinski.

Figures (1)

• Figure 1: Example choices of the functions $\psi(x)$ and $\phi(x)$, plotted using Desmos Desmos. Importantly, each function is smooth, compactly supported, piecewise monotonic and bounded.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1: tuxanidy2024infinitude
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 2.5
• proof : Proof of Theorem \ref{['asymthm']} assuming Proposition \ref{['equiprop1']} and Theorem \ref{['powerthm']}.
• Lemma 3.1
• proof